Zamolodchikov asymptotic formula and instanton expansion in N=2 SUSY N_f=2N_c QCD

The AGT relations allow one to convert the Zamolodchikov asymptotic formula for the conformal block into the instanton expansion of the Seiberg-Witten prepotential for theory with two colors and four fundamental flavors. This provides an explicit formula for the instanton corrections in this model, resolving in this way an old problem in Seiberg-Witten theory. The answer is especially elegant for vanishing matter masses, then the bare charge of gauge theory 16q_0 = 16e^{i\pi\tau_0} plays the role of a branch point on the spectral torus. The exact prepotential at this point is F a^2\log q with q\neq q_0, unlike the case of another conformal theory, with massless adjoint field. Instead, 16q_0 = \theta_{10}^4/\theta_{00}^4(q) = 16q(1+O(q)), i.e. the Zamolodchikov asymptotics gives rise, in particular, to an exact non-perturbative beta-function so that the effective coupling differs from the bare charge by infinite number of instantonic corrections.